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A002415
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4-dimensional pyramidal numbers: n^2*(n^2-1)/12.
(Formerly M4135 N1714)
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78
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0, 0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736, 111265, 124950, 139860, 156066, 173641, 192660, 213200, 235340
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OFFSET
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0,4
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COMMENTS
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Also number of ways to legally insert two pairs of parentheses into a string of m := n-1 letters. (There are initially 2C(m+4,4) (A034827) ways to insert the parentheses, but we must subtract 2(m+1) for illegal clumps of 4 parentheses, 2m(m+1) for clumps of 3 parentheses, C(m+1,2) for 2 clumps of 2 parentheses and (m-1)C(m+1,2) for 1 clump of 2 parentheses, giving m(m+1)^2(m+2)/12 = n^2*(n^2-1)/12.) See also A000217.
E.g. for n=2 there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)), a((b)).
Let M_n denote the n X n matrix M_n(i,j)=(i+j); then the characteristic polynomial of M_n is x^(n-2) * (x^2-A002378(n)*x - a(n)). - Benoit Cloitre, Nov 09 2002
Let M_n denote the n X n matrix M_n(i,j)=(i-j); then the characteristic polynomial of M_n is x^n + a(n)x^(n-2). - Michael Somos, Nov 14 2002
a(n)+1 is the determinant of the n X n matrix M with M(i,i)=1, M(i,j)=i-j. - Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003
Number of permutations of [n] which avoid the pattern 132 and have exactly 2 descents. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
Number of tilings of a <2,n,2> hexagon.
a(n) = number of squares with corners on an n X n grid. See also A024206, A108279.
Kekule numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005
Number of distinct components of the Riemann curvature tensor. - Gene Ward Smith (genewardsmith(AT)gmail.com), Apr 24 2006
Convolution of natural numbers (A001477) with squares (A000290) - Graeme McRae, Jun 06 2006
a(n) is the number of 4 X 4 matrices (symmetrical about each diagonal) M = [a,b,c,d;b,e,f,c;c,f,e,b;d,c,b,a] with a+b+c+d=b+e+f+c=n+2; (a,b,c,d,e,f natural numbers). - Philippe DELEHAM, Apr 11 2007
If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
a(n) = number of Dyck (n+1)-paths with exactly n-1 peaks. - David Callan, Sep 20 2007
Starting (1,6,20,50,...) = third partial sums of binomial transform of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+3,i+3)*b(i)}, where b(i)=[1,2,0,0,0,...]. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
4-dimensional square numbers. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
Equals row sums of triangle A177877; a(n), n>1 = (n-1) terms in (1,2,3,...) dot (...3,2,1) with additive carryovers. Example: a(4) = 20 = (1,2,3) dot (3,2,1) with carryovers = (1*3) + (2*2 + 3) + (3*1 + 7) = (3 + 7 + 10).
Convolution of the triangular numbers A000217 with the odd numbers A004273.
a(n+2) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and w-x=max{w,x,y,z}-min{w,x,y,z}. [Clark Kimberling, May 28 2012]
The second level of finite differences is 1, 4, 9, 16, 25, 36..., the squares. - J. M. Bergot, May 29 2012
Because the differences of this sequence give A000330, this is also the number of squares in an n+1 by n+1 grid whose sides are not parallel to the axes.
a(n+2) gives the number of 2*2 arrays that can be populated with 0..n such that rows and columns are nondecreasing. - Jon Perry, Mar 30 2013
For n consecutive numbers 1,2,3,....n, the sum of all ways of adding the k-tuples of consecutive numbers for n=a(n+1). As an example, let n=4: (1)+(2)+(3)+(4)=10; (1+2)+(2+3)+(3+4)=15; (1+2+3)+(2+3+4)=15; (1+2+3+4)=10 and the sum of these is 50=a(4+1)=a(5). - J. M. Bergot, Apr 19 2013
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REFERENCES
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O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).
Duane DeTemple, "Using Squares to Sum Squares", The College Mathematics Journal, ? (2010), 214-221. [From Parthasarathy Nambi, Jul 16 2010]
R. Euler and J. Sadek, "The Number of Squares on a Geoboard", Journal of Recreational Mathematics, 251-5 30(4) 1999-2000 Baywood Pub. NY
Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration of meanders. Ann. Comb. 6 (2002), no. 1, 7-17.
M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, Arxiv preprint arXiv:1208.1052, 2012. - From N. J. A. Sloane, Dec 24 2012
G. Kreweras, Traitement simultane du "Probleme de Young" et du "Probleme de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle. Institut de Statistique, Universit\'{e} de Paris, 10 (1967), 23-31.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.
P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138. See Table I. - N. J. A. Sloane, Mar 27 2012
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
H. Bottomley, Illustration of initial terms
Milan Janjic, Two Enumerative Functions
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Riemann Tensor.
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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G.f.: x^2*(1+x)/(1-x)^5. [Simon Plouffe in his 1992 dissertation]
a(n) = sum(i=0..n, (n-i)*i^2) = a(n-1)+A000330(n-1) = A000217(n)*A000292(n-2)/n = A000217(n)*A000217(n-1)/3 = A006011(n-1)/3. - Henry Bottomley, Oct 19 2000
a(n) = 2*C(n+2, 4)-C(n+1, 3). - Paul Barry, Mar 04 2003
a(n) = C(n+2, 4)+C(n+3, 4). - Paul Barry, Mar 13 2003
a(n) = sum(k=1, n, sum(i=1, k-1, i^2)). - Benoit Cloitre, Jun 15 2003
a(n) = n*C(n+1,3)/2 = C(n+1,3)*C(n+1,2)/(n+1). - Mitch Harris, Jul 06 2006
a(n) = A006011(n)/3 = A008911(n)/2 = A047928(n-1)/12 = A083374(n-1)/6. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007
a(n) = 1/2*sum {1 <= x_1, x_2 <= n} (det V(x_1,x_2))^2 = 1/2*sum {1 <= i,j <= n} (i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
a(n) = C(n^2,2)/6,n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008
a(n)=C(n+1,3)+2*C(n+1,4). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = (1/48)Sinh[2*ArcCosh[n]]^2. - Artur Jasinski, Feb 10 2010]
a(n) = n*A000292(n-1)/2. - Tom Copeland, Sep 13 2011
a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5), n>4. - Harvey P. Dale, Nov 29 2011]
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MAPLE
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A002415 := proc(n) binomial(n^2, 2)/6 ; end proc: - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008
a:=n->(sum((numbperm(n, 3)), j=2..n)):seq(a(n)/12, n=1..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2008
a:=n->(sum((numbcomp(n, 4)), j=3..n))/2:seq(a(n), n=2..40); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
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MATHEMATICA
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Table[(n^4 -n^2 )/12, {n, 0, 40}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007
Table[(1/48) Round[N[Sinh[2 ArcCosh[n]]^2, 100]], {n, 0, 50}] (*Artur Jasinski*) [From Artur Jasinski, Feb 10 2010]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 1, 6, 20}, 40] (* Harvey P. Dale, Nov 29 2011 *)
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PROG
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(PARI) a(n)=n^2*(n^2-1)/12
(PARI) a(n)=sum(k=1, n, sum(m=1, k, sum(i=1, m, (2*i-1)))) - Alexander R. Povolotsky, Nov 05 2007
sage: [lucas_number1(3, n^2, n^2)/12 for n in xrange(0, 39)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008
(JavaScript)
for (n=1; n<50; n++) {
x=0;
for (a=0; a<n; a++)
for (b=0; b<n; b++)
for (c=0; c<n; c++)
for (d=0; d<n; d++)
if (a<=b && a<=c && b<=d && c<=d) x++;
document.write(x+", ");
}
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CROSSREFS
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a(n)= ((-1)^n)*A053120(2*n, 4)/8 (one eighth of fifth unsigned column of Chebyshev T-triangle, zeros omitted). Cf. A001296.
Second row of array A103905.
Third column of Narayana numbers A001263.
Cf. A001079, A002415, A006011, A006542, A008911, A037270, A047819, A047928, A071253, A083374, A107891, A108741, A132592, A146311, A146312, A146313, A173115, A173116.
Partial sums of A000330.
The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers (A000027) with the k-gonal numbers.
Sequence in context: A161699 A216175 A161409 * A052515 A067117 A213586
Adjacent sequences: A002412 A002413 A002414 * A002416 A002417 A002418
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Oct 19 2000
Typo in link fixed by Matthew Vandermast, Nov 22 2010
Redundant comment deleted and more detail on relationship with A000330 added by Joshua Zucker, Jan 01 2013
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STATUS
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approved
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